95] NEWTONIAN POTENTIAL FUNCTION. 185 



where Y n is a homogeneous function of the trigonometric functions 

 cos 6, sin 6 cos <, and sin sin <f>. T n being the value of V n on the 

 surface of a sphere of unit radius, is called a surface harmonic. 

 The equation Y n = represents a cone of order n, whose inter- 

 section with the sphere gives a geometrical representation of the 

 harmonic V n . 



If u and v be any two continuous functions of x, y, z, 



a 2 (uv) = u ^v , 2 + 



9# 2 da? dx dx da? ' 



/ \ A / \ A A , o fi u d y du dv du dv 



(3) A (uv) = u&v + v&u + 2^-3-+ + 5- 



\dx dx dy dy dz dz 



Put u = r m , and since 



_vU __ mr m-i ^ 



ox ox 



02 ( r m\ 



} 



we get 



(4) A (r m ) = 3rar m ~ 2 + m (m - 2) r m 



= m (m + 1) r m ~ 2 . 

 If V n is a harmonic of degree n, 



(5) A (r m V n ) = r m A F n + m (m + 1) r m ~ 2 F n 



= [m (m + 1) + 2m^] r m ~ 2 V n , 

 by virtue of equations (i) and (2). 



Consequently if m = (2w + l), the product r m F w is a harmonic. 

 Since V n is of degree w, and r is of degree unity in the coordinates, 

 r -(2n+i) Y n is of degree (n + 1). Accordingly to any spherical 

 harmonic V n = r n Y n of degree n there corresponds another, 



v Vn 



-(n+i) ^+i> 



of degree (n + 1). Compare this with the corresponding property 

 of circular harmonics, where the degrees of the two corresponding 

 harmonics are equal and opposite. 



